A model-based opinion dynamics approach to tackle vaccine hesitancy

Uncovering the mechanisms underlying the diffusion of vaccine hesitancy is crucial in fighting epidemic spreading. Toward this ambitious goal, we treat vaccine hesitancy as an opinion, whose diffusion in a social group can be shaped over time by the influence of personal beliefs, social pressure, and other exogenous actions, such as pro-vaccine campaigns. We propose a simple mathematical model that, calibrated on survey data, can predict the modification of the pre-existing individual willingness to be vaccinated and estimate the fraction of a population that is expected to adhere to an immunization program. This work paves the way for enabling tools from network control towards the simulation of different intervention plans and the design of more effective targeted pro-vaccine campaigns. Compared to traditional mass media alternatives, these model-based campaigns can exploit the structural properties of social networks to provide a potentially pivotal advantage in epidemic mitigation.

whose expected value and variance can be computed as respectively.
Note that Y i is a Poisson binomial distribution (scaled by the factor 1/n), that is the sum of n independent Bernoulli distributions. Here, we study how its moments scale with the population size n. Denoting n 0 = 1446 the number of participants to the survey on which the opinion dynamics model is parametrized in the main text, we scale the population as multiples of n 0 , so that we can always associate a vaccination probability p i to a fraction 1/n 0 of the total population n, for all i = 1, . . . , n 0 . Specifically, we introduce the parameter β k := l k /n 0 with l k = 10 k , and k = 4, . . . , 7. This gimmick allows us to inspect the behaviour of the moments of the Poisson Binomial distribution when the size of the population is n = β k n 0 . In turn, from equation (S1) this reflects into the following scaling behavior of the first and second moment of Y n Hence, • the expected value E[Y n ] of the fraction of the population that will get a shot of vaccine does not change with the population size; • the variance Var[Y n ] decreases linearly with the population size. Table S1 reports the mean and variance of Y n for different orders of magnitudes for l k , whereas Figure S1 the error bar of the fraction of vaccinated population as a function of the population size n.  Table S1: First and second moment of Y n as the population size n = β k n 0 varies. Figure S1: Error bar plot of the mean and standard deviation of Y n .

S2 Parametric analysis of ρ
In the second subsection of the Methods section of the main text, we presented a constrained least square optimization problem aimed to realistically calibrate the model parameters consistently with survey data. In particular, in the in the main text we explained that constraint (4c) sets the average susceptibility to be equal to a value ρ, and that problem (4) admits a solution only if ρ≤0.58. In the main text, all the analysis have been performed for ρ = 0.58. Here, we perform a parametric analysis of the results whereby we vary ρ in the interval [0.18, 0.48] with step 0.1. Figure S2 illustrates that the results are qualitatively similar, the only difference being the attenuated effectiveness of all the strategies, since lower values of the λ i correspond to a more stubborn population. Figure S2: Comparison of targeted and traditional mass campaigns for different values of the average susceptibility ρ (set to 0.18, 0.28, 0.38, and 0.48, from top to bottom). The left panels depict, for each effort η, the additional population fraction ∆ µ and ∆ 0 µ that is expected to be vaccinated when the best targeted campaign (identified by circles) or the mass campaign (identified by triangles) are employed, respectively. The right panels display for each effort η and targeted strategy s, the ratio between the fractions of the population µ s (η) and µ 0 (η) that are expected to be vaccinated when strategy s and the traditional campaign are employed, respectively. In all panels, Strategies 1, 2, 3 are depicted in blue, green, and magenta, respectively, and the intensity of the vaccination campaign is set to α = 1.

S3 Robustness analysis
To test the robustness of our results we have run additional simulations varying the graph through which opinions diffuse. To do so, we repeated the same analysis performed in the main text on a pool of real and synthetic networks, whereby we compared the effectiveness of the targeted campaigns to that of mass campaigns. As shown in Figures S3-S6, the results are consistent with the case illustrated in the main text, that is, the targeted campaigns always outperforms the traditional mass-media ones.
The pool of synthetic networks is made of 10 unweighted undirected graphs of size N = 1500, extracted from a Scale-Free distribution with exponent γ = 2.8 and average degree k av = 80, consistently with the properties of the real online social networks reported in [1], and repeated the same analysis performed in the main text. The 3 real networks, called soc-fbHamilton46, socfb-Simmons81 and Hamsterster, have been retrieved from the network repository [1]. Figure S3: Comparison of targeted and traditional mass campaigns effectiveness averaged on a pool of 10 Scale-Free synthetic networks. The left panels depict, for each effort η, the additional population fraction ∆ µ and ∆ 0 µ that is expected to be vaccinated when the best targeted campaign (identified by circles) or the mass campaign (identified by triangles) are employed, respectively. The right panels display for each effort η and targeted strategy s, the ratio between the fractions of the population µ s (η) and µ 0 (η) that are expected to be vaccinated when strategy s and the traditional campaign are employed, respectively. In all panels, Strategies 1, 2, 3 are depicted in blue, green, and magenta, respectively, and the intensity of the vaccination campaign is set to α = 1.

S4 Supplementary Figure
In Figure S7, we report a graphical representation of the calibration procedure described in the Methods Section of the main text. Figure S7: Violin plots of the steady-state opinion distribution from the model calibrated as described in the Methods (light blue), with that obtained from the survey data (blue), respectively. On the right, the violin plots of the initial and final opinions distribution of vaccine willingness, respectively. Data points corresponding to agents' opinions in the two endpoints are colored accordingly to their Likert score on vaccine willingness survey.